Grazzt
Demon Lord
One of the most interesting that I've seen is the HackMaster 5th Edition system, which uses a "count up" system. There is no round - time is measured in seconds, and actions you decide to take use up so many seconds. Swing a sword? You can do that once every 10 seconds. Swing a two-handed sword? You can do that once every 18 seconds. Etc., etc. You can move 5 feet in one second, but then you can jog 10 feet the next second, and then break into a run on the third second. It's pretty intuitive BUT it's a huge pain for the GM I've found and almost totally reliant on a grid & miniatures.
Aces & Eights (also by Kenzer) uses this system. Each combatant rolls initiative. Modified by speed. Lower is better because the gamemaster starts the "count up" and when your count is reached, you can start to act. Say you roll and after modifying for Speed your count is 3. When the count reaches 3 you act. If you have your gun drawn, you can fire (1 count). If you have to draw your gun, it's like 5 counts, so you next act on 8. At 8 count, you can fire from hip or bring gun to bear and aim (like 4 counts). So at count 12, you're aiming and can fire. Cocking the hammer and firing again is like 3 or 4 counts, so you can fire again on count 15 or 16. Meanwhile, the other guy(s) are acting on their counts.
It works well with a very small number of guys fighting. Otherwise it gets to be a bit much when there's a lot of combatants.